You wrote $\displaystyle u=\left(\frac{2p_2x_2}{p_1}\right)^2\!\!x_2=\left( \frac{2p_2}{p_1}\right)^2\!\!x_2^2x_2=\left(\frac{ 2p_2}{p_1}\right)^2\!\!x_2^3\Longrightarrow u^{1\slash 3}=\left(\frac{2p_2}{p_1}\right)^{2\slash 3}\!\!x_2$ ...and from here isolate $\displaystyle x_2$ and get the result.
Tonio
Hello, matlondon!
$\displaystyle U \;=\;\left(\frac{2P_2x_2}{P_1}\right)^2x_2$
$\displaystyle x_2 \;=\;U^{\frac{1}{3}}\left(\frac{P_1}{2P_2}\right)^ {\frac{2}{3}} $
How do i get fron $\displaystyle U$ to $\displaystyle x_2$ ?
We have: . $\displaystyle \left(\frac{2P_1x_2}{P_1}\right)^2x_2 \;=\;U $
. . . . . $\displaystyle \left(\frac{2P_2}{P_1}\right)^2(x_2)^2 \cdot x_2 \;=\;U$
. . . . . . . $\displaystyle \left(\frac{2P_2}{P_1}\right)^2(x_2)^3 \;=\;U $
. . . . . . . . . . . . $\displaystyle (x_2)^3 \;=\;U\left(\frac{P_1}{2P_2}\right)^2 $
. . . . . . . . . . . . . .$\displaystyle x_2 \;=\;\left[U\left(\frac{P_1}{2P_2}\right)^2\right]^{\frac{1}{3}} $
. . . . . . . . . . . . . .$\displaystyle x_2 \;=\;U^{\frac{1}{3}}\left(\frac{P_1}{2P_2}\right)^ {\frac{2}{3}} $
Because that's what has to be done when passing to the other side of an equation something that was multiplying: if $\displaystyle A\neq 0\,\,\,and\,\,\,Ax=Y\,\,\,then\,\,\,x=\frac{Y}{A}$ . If A happens to be a fraction then when it passes to the other side dividing the fraction "flips". It's simply fractions arithmetic.
Tonio